Bring a large pot of salted water to a boil, add the pasta, and cook for 8 to 10 minutes, to the point that the pasta is aldente. Drain the pasta well, reserving at least ½ cup of the starchy cooking water to use in the sauce.
Meanwhile, heat 2 tablespoons of the olive oil in a deep skillet over medium heat. Add the oven-dried tomatoes to the pan, increase the heat to high, and cook for about 2 minutes, until the tomatoes are slightly caramelized. Turn the heat down to medium, add the garlic and sweet peas, and cook for another 2 minutes.
Beat the eggs and cheese together in a medium bowl, stirring well to prevent lumps. Add the hot, drained pasta to the pan along with the remaining 1 tablespoon olive oil and toss to coat the pasta. Reduce the heat to maintain a simmer and pour the egg mixture into the pasta, mixing quickly until the eggs thicken but do not scramble (it is lower risk to do this off the heat, but the results will never be as good). Thin out the sauce with a bit of the reserved pasta water, until it just clings to the pasta. Taste the dish and adjust the seasoning.
Divide the carbonara among plates and garnish with the parsley.
© Modern Kosherby Michael Aaron Gardiner, Rizzoli New York, 2020